Sharp edges not sharp enough when bending?
You're using NGons (faces with more than 4 vertices/edges) and a curve converted to mesh, resulting in bad triangulated topology.
They're exactly what you don't want to bend.

Smoothed curve surfaces need quad topology.

All you can do is trying the face beauty fill (Ctrl+F menu with faces selected), or selecting the flat faces deleting them and reconstruct something better (extruding/scaling inside the outer loop) or using remesh modifier.

For your big round face of the background, inset it with I multiple times to the center and Alt+M at the center to merge the last loop at the center.

This is an exemple of a manual retopo. All quads, can be subsurfed, no bending artifact.

Blend file here